The commutator of two elements, g and h, of a groupG, is the element
[g, h] = g−1h−1gh.
It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of Ggenerated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups.
The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as
Commutator identities are an important tool in group theory. The expression ax denotes the conjugate of a by x, defined as x−1ax.
Identity (5) is also known as the Hall–Witt identity, after Philip Hall and Ernst Witt. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).
N.B., the above definition of the conjugate of a by x is used by some group theorists. Many other group theorists define the conjugate of a by x as xax−1. This is often written . Similar identities hold for these conventions.
Many identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well:
It is zero if and only if a and b commute. In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.
The anticommutator of two elements a and b of a ring or an associative algebra is defined by
Sometimes the brackets [ ]+ are also used to denote anticommutators, while [ ]− is then used for commutators. The anticommutator is used less often than the commutator, but can be used for example to define Clifford algebras, Jordan algebras and is utilised to derive the Dirac equation in particle physics.
An additional identity may be found for this last expression, in the form:
If A is a fixed element of a ring R, the first additional identity can be interpreted as a Leibniz rule for the map given by . In other words, the map adA defines a derivation on the ring R. The second and third identities represent Leibniz rules for more than two factors that are valid for any derivation. Identities 4–6 can also be interpreted as Leibniz rules for a certain derivation.
McKenzie, R.; Snow, J. (2005), "Congruence modular varieties: commutator theory", in Kudryavtsev, V. B.; Rosenberg, I. G., Structural Theory of Automata, Semigroups, and Universal Algebra, Springer, pp. 273–329